PPT: Z-Transform & Region of Convergence | Signals and Systems - Electrical Engineering (EE) PDF Download

 Page 1


1
The z-Transform
Page 2


1
The z-Transform
2
z-Transform
§ The z-transform is the most general concept for the
transformation of discrete-time series.
§ The Laplace transform is the more general concept for
the transformation of continuous time processes.
§ For example, the Laplace transform allows you to
transform a differential equation, and its corresponding
initial and boundary value problems, into a space in
which the equation can be solved by ordinary algebra.
§ The switching of spaces to transform calculus problems
into algebraic operations on transforms is called
operational calculus. The Laplace and z transforms are
the most important methods for this purpose.
Page 3


1
The z-Transform
2
z-Transform
§ The z-transform is the most general concept for the
transformation of discrete-time series.
§ The Laplace transform is the more general concept for
the transformation of continuous time processes.
§ For example, the Laplace transform allows you to
transform a differential equation, and its corresponding
initial and boundary value problems, into a space in
which the equation can be solved by ordinary algebra.
§ The switching of spaces to transform calculus problems
into algebraic operations on transforms is called
operational calculus. The Laplace and z transforms are
the most important methods for this purpose.
3
The Transforms
The Laplace transform of a function f(t):
ò
¥
-
=
0
) ( ) ( dt e t f s F
st
The one-sided z-transform of a function x(n):
å
¥
=
-
=
0
) ( ) (
n
n
z n x z X
The two-sided z-transform of a function x(n):
å
¥
-¥ =
-
=
n
n
z n x z X ) ( ) (
Page 4


1
The z-Transform
2
z-Transform
§ The z-transform is the most general concept for the
transformation of discrete-time series.
§ The Laplace transform is the more general concept for
the transformation of continuous time processes.
§ For example, the Laplace transform allows you to
transform a differential equation, and its corresponding
initial and boundary value problems, into a space in
which the equation can be solved by ordinary algebra.
§ The switching of spaces to transform calculus problems
into algebraic operations on transforms is called
operational calculus. The Laplace and z transforms are
the most important methods for this purpose.
3
The Transforms
The Laplace transform of a function f(t):
ò
¥
-
=
0
) ( ) ( dt e t f s F
st
The one-sided z-transform of a function x(n):
å
¥
=
-
=
0
) ( ) (
n
n
z n x z X
The two-sided z-transform of a function x(n):
å
¥
-¥ =
-
=
n
n
z n x z X ) ( ) (
4
Relationship to Fourier Transform
Note that expressing the complex variable z in
polar form reveals the relationship to the
Fourier transform:
å
å
å
¥
-¥ =
-
¥
-¥ =
- -
-
¥
-¥ =
= =
= =
=
n
n i i
n
n i n i
n
n
i i
e n x X e X
r if and e r n x re X
or re n x re X
w w
w w
w w
w ) ( ) ( ) (
, 1 , ) ( ) (
, ) )( ( ) (
which is the Fourier transform of x(n).
Page 5


1
The z-Transform
2
z-Transform
§ The z-transform is the most general concept for the
transformation of discrete-time series.
§ The Laplace transform is the more general concept for
the transformation of continuous time processes.
§ For example, the Laplace transform allows you to
transform a differential equation, and its corresponding
initial and boundary value problems, into a space in
which the equation can be solved by ordinary algebra.
§ The switching of spaces to transform calculus problems
into algebraic operations on transforms is called
operational calculus. The Laplace and z transforms are
the most important methods for this purpose.
3
The Transforms
The Laplace transform of a function f(t):
ò
¥
-
=
0
) ( ) ( dt e t f s F
st
The one-sided z-transform of a function x(n):
å
¥
=
-
=
0
) ( ) (
n
n
z n x z X
The two-sided z-transform of a function x(n):
å
¥
-¥ =
-
=
n
n
z n x z X ) ( ) (
4
Relationship to Fourier Transform
Note that expressing the complex variable z in
polar form reveals the relationship to the
Fourier transform:
å
å
å
¥
-¥ =
-
¥
-¥ =
- -
-
¥
-¥ =
= =
= =
=
n
n i i
n
n i n i
n
n
i i
e n x X e X
r if and e r n x re X
or re n x re X
w w
w w
w w
w ) ( ) ( ) (
, 1 , ) ( ) (
, ) )( ( ) (
which is the Fourier transform of x(n).
5
Region of Convergence
The z-transform of x(n) can be viewed as the Fourier
transform of x(n) multiplied by an exponential
sequence r
-n
, and the z-transform may converge
even when the Fourier transform does not.
By redefining convergence, it is possible that the
Fourier transform may converge when the z-
transform does not.
For the Fourier transform to converge, the
sequence must have finite energy, or:
¥ <
å
¥
-¥ =
-
n
n
r n x ) (